Let \(\pi\colon \mathbb{Z}\times \mathbb{Z}\to \mathbb{Z}\) be given by \(\pi(a,b)=a\). To which ring is \((\mathbb{Z}\times \mathbb{Z})/\ker\pi\) isomorphic?
Since \(\pi\) is a surjective homomorphism, the first theorem says the quotient is isomorphic to the image, which is all of \(\mathbb{Z}\). Here \(\ker\pi=\{0\}\times \mathbb{Z}\).
If \(\varphi\colon R\to S\) is a ring homomorphism with kernel \(\ker\varphi\), the quotient ring of \(R\) by \(\ker\varphi\) is isomorphic to the image of \(\varphi\).
\[R/\ker\varphi\;\cong\;\varphi(R)\]
More practice problems: first isomorphism theorem for rings